Similarly, the estimands \(\phi \)  and \(\xi \)  may be compared to \(DE(\alpha )\)  and \(IE_0(\alpha ,\alpha ^{\prime })\) , which are discussed in [1]}, [2]}, and are given by
\(DE(\alpha ) &= \frac{1}{N}\sum _{i=1}^N \bigg (\mathbb {E}[Y_i | X_i = 1; \alpha ] - \mathbb {E}[Y_i | X_i=0; \alpha ]\bigg ) \\IE_0(\alpha ,\alpha ^{\prime }) &= \frac{1}{N} \sum _{i=1}^N \bigg ( \mathbb {E}[Y_i | X_i = 0; \alpha ] - \mathbb {E}[Y_i| X_i =0; \alpha ^{\prime }]\bigg )\) 
